Disease Localization in Multilayer Networks.

G. Ferraz de Arruda, E. Cozzo, T. P. Peixoto, F. A. Rodrigues, and Y. Moreno, *“Disease Localization in Multilayer Networks”*, ** Physical Review X** **7**, 011014 (2017).

Current modeling of infectious diseases allows for the study of complex and realistic scenarios that go from the population to the individual level of description.

*real demographic distributions, detailed migration and mobility patterns, human behavioral responses to the presence of an outbreak, disease’s multi-strain, and the concurrency of several diseases, among others*. Most epidemic models assume that the spreading process takes place on a single level

*(be it a single population, a meta-population system or a network of contacts)*. The latter results from our current limited knowledge about the interplay among the various scales involved in the transmission of infectious diseases at the global scale. Therefore, pressing problems rooted at the interdependency of multi-scales call for the development of a whole new set of theoretical and simulation approaches. It is extremely importatnt to develop disease-specific theoretical and computational models to understand the transmission mechanisms behind global public health threats. The fact of considering the interdependency between many levels and scales represents a radical change that makes a substantial difference as it allows addressing problems such as comorbidity and the spreading of persistent infections and multi-strain diseases. Thus, the goal consists of developing a contemporary epidemiological framework that integrates the many aspects involved in the spreading of global diseases: from single networks of contacts to meta-population systems and from the interaction between different diseases and strains to the influence of human behavioral changes and mobility patterns.

The description of the evolution of a single disease on a particular network of contacts has been the subject of intense research during the last several years. Nowadays, we have a set of tools that allows addressing the previous challenge to a large degree of details. However, in many situations, multiple pathogens coexist within the same host population and usually interact among each other. This includes, for example, systems of competing pathogens (e.g., seasonal influenza) or the so-called syndemic systems (e.g. HIV and Tuberculosis), i.e., two pathogens each of which enhances or impairs the spreading of the other.

In this paper, we develop a theoretical and computational framework to study the dynamics of concurrent diseases. We propose a model for the description of the simultaneous spreading of two interacting pathogens on the same host population through independent contact networks. The model is based on an heterogenous mean-field approach to describe the critical properties of the dynamics as well as an adequate framework for the temporal description of coupled out of equilibrium outbreaks for both the Susceptible-Infected-Susceptible and Susceptible-Infected-Removed scenarios. The proposed framework allows to analytically derive the epidemic thresholds of the diseases modeled, explicitly addressing the influence on each threshold on aspects such as the prevalence of the conjugate disease, the system size, the architecture of the networks of contacts and the appearance of eventual correlations between them. Overall, our findings provide deep insights into what are the key mechanisms that drive the evolution of interacting diseases and secondly, they pave the way for the development of quantitative, data-driven models for the detailed characterization of concurrent and interacting diseases.

J. Sanz, C.-Y. Xia, S. Meloni and Y. Moreno, *“Dynamics of interacting diseases”*, **Physical Review X** **4**, 041005 (2014).

The inclusion of mobility processes is a key ingredient in the modeling of the geographic spread of epidemics. Models that explicitly take into account the mobility patterns of individuals range from relatively coarse-grained approaches that consider aggregated traveling flows to highly detailed structured meta-population or agent-based models allowing for the description of billions of individuals.

S. Meloni, N. Perra, A. Arenas, S. Gomez, Y. Moreno, and A. Vespignani, *“Modeling Human Mobility Responses to the Large-scale Spreading of Infectious Diseases”*, **Scientific Reports** **1**, 62 (2011).

Current state-of-the-art epidemiological models incorporating complex patterns of interactions (no matter whether at the level of meta-populations or not) are devised to deal with a single spreading process, i.e., *they are not thought to deal with situations in which several strains of the same disease coexist.*

C. Poletto, S. Meloni, V. Colizza, Y. Moreno and A. Vespignani, *“Host mobility drives pathogen competition in spatially structured populations”*, **PLoS Computational Biology** **9** (8): e1003169 (2013).

C. Poletto, S. Meloni, A. Van Metre, V. Colizza, Y. Moreno and A. Vespignani, *“Characterizing two-pathogen competition in spatially structured environments”*, **Scientific Reports** **5**:7895 (2015).

The critical properties of an epidemic outbreak in complex networks were first addressed using the heterogeneous mean-field (HMF) prescription. This framework has been proved to be exact in annealed networks, whose nodes’ degrees are sampled from a fixed degree distribution at each step of the dynamics (*i.e. its specific connectivity is fixed only in average*).

*quenched networks*), HMF can result in different levels of accuracy. This problem leads to the question of whether or not the direct use of the HMF approach is accurate enough when dealing with real networks. On the other hand, meta-population approaches are an essential theoretical framework used in Epidemiology, population ecology, genetics and adaptive evolution to describe population dynamics whenever the spatial structure of populations plays an important role in the system’s evolution. The basic assumption of meta-population models is that the system under study is highly fragmented and characterized by populations localized in relatively isolated discrete units/subpopulations connected by some degree of migration and or commuting. Two different dynamics take place concurrently: inside each subpopulation and between subpopulations. The most important difference with respect to single population models is the existence of a second epidemic threshold. This new critical point, known as global invasion threshold, marks the point beyond which a local outbreak reaches other subpopulations and spreads throughout the meta-population system. The global invasion point does not only depend on the infection parameters, but also on the mobility rates of individuals as well and thus differs from the single population epidemic threshold. Recent works have extensively applied meta-population schemes to understand the epidemic dynamics of spatially structured populations. However, theoretical approaches are all at the mean-field level and use a tree-like approximation that represents the evolution of the number of diseased subpopulations as a branching process. In addition, individuals within the subpopulations are all well mixed and no sources of heterogeneity (neither in the networks of contacts nor in the individual mobility rates) at the lower individual scale are built in the models. Finally, data-driven computations are very intensive and lengthy, especially when modeling disease propagation worldwide. We therefore aim at integrating individual-level approaches into meta-population schemes to add further realism to the structure of subpopulations.

**© June 2013, COSNET Lab**

**·**BIFI, Zaragoza, Spain. All rights reserved.